Integrals giving inverse trig functions
Problem 4.636 · medium
Evaluate \( \displaystyle \int \frac{4}{x^{2} - 6 x + 13}\, dx \).
- \[ x^{2} - 6 x + 13 = \left(x - 3\right)^{2} + 4 \]Complete the square.✓ Proved
- With u = x − 3 and a = 2, this is c·∫ du/(u² + a²)
- \[ \frac{d}{d x} 2 \operatorname{atan}{\left(\frac{x}{2} - \frac{3}{2} \right)} = \frac{4}{x^{2} - 6 x + 13} \]An antiderivative is 2*atan(x/2 - 3/2); differentiate to confirm.✓ Proved
Answer \( 2 \operatorname{atan}{\left(\frac{x}{2} - \frac{3}{2} \right)} + C \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | difference quotient of the answer at x = -4.0 |
Reviewers
gpt-oss:20b: fail (error) — The sentence claims the integral reduces to a constant times ∫du/(u²+a²) but omits the crucial factor 4 that comes from the numerator. This omission would mislead a student into thinking the antiderivative is 2·atan((x−3)/2) instead of the correct 2·atan((x−3)/2).qwen3.6:27b-mlx: pass — The solution correctly completes the square, identifies the standard integral form, and verifies the result by differentiation. The steps are logically sound and the final answer is correct.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — The sentence claims the integral reduces to a constant times ∫du/(u²+a²) but omits the crucial factor 4 that comes from the numerator. This omission would mislead a student into thinking the antiderivative is 2·atan((x−3)/2) instead of the correct 2·atan((x−3)/2).qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly completes the square, identifies the standard integral form, and verifies the result by differentiation. The steps are logically sound and the final answer is correct.gpt-oss:20b: fail (misleading) 2026-10-07 — The sentence "With u = x − 3 and a = 2, this is c·∫ du/(u² + a²)" omits the crucial constant factor. The integrand becomes 4∫du/(u²+4), not just a constant times the standard form; the missing factor 4 leads to an incorrect intermediate statement and could mislead a student about the role of the constant multiplier.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly completes the square and identifies the standard arctangent integral form. The final verification by differentiation confirms the result is correct.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/inverse_trig_integral, checked 2026-10-07 with SymPy 1.14.0.